Spectrophotometer is one of the most widely used analytical instruments to characterize materials in research, development and industrial applications. Existing spectrophotometers, however, lack the ability to accurately characterize materials with different types of turbidity. A material is defined as turbid if it exhibits light scattering in which the scattered light has the same wavelength as that of the incident light. Many natural and artificial materials are of different degrees of turbidity which include polluted water, contaminated oil, paint, milk, blood, aqueous suspensions of biological cells and/or nanoparticles, biological and human tissues. Many of these materials exhibit strong light scattering characteristics, in addition to light absorption, in their interaction with light in the optical spectrum from ultraviolet to infrared. One of the most accurate optical models to characterize light absorption and scattering in turbid materials is provided by combining the radiative transfer (RT) theory to describe light transportation in a turbid material and the Fresnel equation to describe light transportation between turbid materials of different real refractive indices.
The RT theory defines an absorption coefficient μa, scattering coefficient μs and a scattering phase function p(s, s′) as the optical parameters characterizing a turbid material, where s and s′ are unit vectors representing the light propagation directions before and after a scattering event. To form a boundary-value problem describing light-material interaction, the RT theory has to be supplemented by proper boundary conditions. Reasonable boundary conditions can be formulated with the Fresnel equation in which the light transportation through an interface between two neighboring materials with mismatched real refractive indices n is treated as transverse electromagnetic wavefields. Furthermore, a Henyey-Greenstein (HG) function p(cos α) has been widely used to represent the scattering phase function in turbid materials, where α is the angle between s and s′ directions and cos α=s·s′. The form of the HG function is fully determined by a single parameter g, where g is called as the anisotropy factor and defined as the mean value of cos α, i.e., g=<cos α>. With the HG function p(cos α) as the scattering phase function, optical characterization of a turbid sample within the framework of the RT theory and Fresnel equation is reduced to the determination of four optical parameters: μa, μs, g and n. In general, spectrophotometric characterization of a turbid material is accomplished by determining these parameters as functions of wavelength in the spectrum of interest, which requires accurate measurement of optical signals from the turbid material sample and accurate calculation of these signals based on the RT theory and Fresnel equation.
In existing spectrophotometers, an incident light intensity signal I0 and a collimated transmitted light intensity signal Ic are measured from a sample with photodetectors. A collimated transmittance signal defined as Tc=Ic/I0 is obtained and used to determine the sample's absorbance A and/or attenuation coefficient μt as a function of wavelength based on the Beer's law (also called as the Beer-Lambert's law). The Beer's law can be derived from RT theory if the scattered light from the sample is not present or can be neglected in the detected signal Ic. The Beer's law states that the collimated transmittance Tc is related to an attenuation coefficient μt as Tc=e−μtD, where μt=μa+μs is the sum of the absorption coefficient μa and scattering coefficient μs and D is the sample thickness along the transmitted light direction. This allows the determination of the absorbance from A=−log10(Tc) or the attenuation coefficient from
      μ    t    =            -              2.30        D              ⁢                  log        10            ⁡              (                  T          c                )            as functions of wavelength λ in the spectrum of interest. We note here that the reflection loss of the light beams at the interfaces between the air and sample holder and between the sample holder and sample is neglected in the definition of Tc.
Characterization of a sample in existing spectrophotometers with either A(λ) or μt(λ) is accomplished with a monochromatic incident beam of adjustable wavelength and single photodetectors for signal measurement in the spectrum of interest. Spectrophotometric characterization can also be accomplished with a broadband incident light beam in the spectrum of interest and appropriate spectral dispersive devices combined with imaging photodetectors for signal measurement. Here, single photodetectors refer to those light detectors such as photodiodes or photomultipliers with one detecting element and one output signal, and the output signal is related to the total light intensity over the area of the detecting element. Imaging photodetectors refer to those light detectors with multiple detecting elements and multiple signal outputs, and each output signal is related to the light intensity over the area of a specific detecting element. Examples of imaging photodetectors include linear array photodetectors and charge-coupled devices.
It is clear from the Beer's law that either A or μt provides only the information on how much light is attenuated in the material but not on the pathways of light attenuation since attenuation can be caused by absorption (μt is due μa) or scattering (μt is due to μs). Furthermore, the existing spectrophotometers have no capability to distinguish samples of same μa and μs but exhibiting different characteristics of light scattering (forward scattering, side scattering and backscattering). As an extreme example, both milk and ink attenuate light strongly with the former mainly through light scattering (so it appears white with μt≈μs) while the latter mainly through light absorption (so it appears dark with μt≈μa). Another example is the investigation of certain paints with strong light backscattering capability which are preferred for making road signs easier to be seen by drivers of automobiles with headlights illuminating the signs. Yet, another example is to distinguish different biological cells without the need to stain them with fluorescence dyes by the spectrophotometric determination of all the optical parameters of the cell suspension samples. These examples illustrate the needs for a new type of spectrophotometers to accurately characterize turbid materials with the optical parameters of μa, μs, g and n.
Determination of the above optical parameters requires accurate measurement of light signals scattered out of a turbid material sample in addition to the collimated transmitted light signal followed by calculation of these signals on the basis of an accurate optical model such as the RT theory and Fresnel equation. Several methods have been developed to determine some of the four optical parameters of μa, μs, g and n. None of these methods, however, can be used to determine μa, μs, g and n in one instrument. Recently, an integrating sphere based method has been developed as a primary method to determine μa, μs, g and n as functions of wavelength. In this method, a device of integrating sphere is used to measure the diffuse reflectance signal Rd and diffuse transmittance signal Td, a spatial filtering device is used to measure the collimated transmittance signal Tc and a prism based device is used to determine the coherent reflectance signal Rc as a function of incident angle θ of a monochromatic light beam. The real refractive index n is obtained by fitting the calculated values of Rc(θ) using the Fresnel equation to the measured values of Rc(θ). This is followed by the determination of the optical parameters of μa, μs and g from the measured signals of Tc, Td and Rd using the Beer's law and a Monte Carlo simulation method within the framework of RT theory.
Despite its ability to determine the optical parameters of μaμs, g and n, the integrating sphere based method requires the use of three optical devices as discussed in the forgoing. The experimental procedures of the integrating sphere based method require that the turbid material be made into slab samples of different thickness (or in sample holders of different thickness) and moved between the three devices. Therefore, this method is time consuming, prone to error and of limited usefulness to industrial and/or biomedical applications.
In view of the forgoing, it would be an advance in the art to provide method and apparatus for measuring optical signals in one instrument from a turbid material sample without the need for an integrating sphere and prism and for determining the optical parameters of a, μs, g and n as functions of wavelength. It would also be an advance in the art to provide the methods and apparatus for accurate and rapid measurement of the optical signals and determination of the optical parameters.